Sufficient Conditions for Apparent Horizons in Spherically Symmetric Initial Data

We establish sufficient conditions for the appearance of apparent horizons in spherically symmetric initial data when spacetime is foliated extrinsically. Let $M$ and $P$ be respectively the total material energy and the total material current contained in some ball of radius $\ell$. Suppose that the dominant energy condition is satisfied. We show that if $M- P \ge \ell$ then the region must possess a future apparent horizon for some non -trivial closed subset of such gauges. The same inequality holds on a larger subset of gauges but with a larger constant of proportionality which depends weakly on the gauge. This work extends substantially both our joint work on moment of time symmetry initial data as well as the work of Bizon, Malec and \'O Murchadha on a maximal slice.Comment: 16 pages, revtex, to appear in Phys. Rev.

Geometric Bounds in Spherically Symmetric General Relativity

We exploit an arbitrary extrinsic time foliation of spacetime to solve the constraints in spherically symmetric general relativity. Among such foliations there is a one parameter family, linear and homogeneous in the extrinsic curvature, which permit the momentum constraint to be solved exactly. This family includes, as special cases, the extrinsic time gauges that have been exploited in the past. These foliations have the property that the extrinsic curvature is spacelike with respect to the the spherically symmetric superspace metric. What is remarkable is that the linearity can be relaxed at no essential extra cost which permits us to isolate a large non - pathological dense subset of all extrinsic time foliations. We identify properties of solutions which are independent of the particular foliation within this subset. When the geometry is regular, we can place spatially invariant numerical bounds on the values of both the spatial and the temporal gradients of the scalar areal radius, $R$. These bounds are entirely independent of the particular gauge and of the magnitude of the sources. When singularities occur, we demonstrate that the geometry behaves in a universal way in the neighborhood of the singularity.Comment: 16 pages, revtex, submitted to Phys. Rev.

Necessary Conditions for Apparent Horizons and Singularities in Spherically Symmetric Initial Data

We establish necessary conditions for the appearance of both apparent horizons and singularities in the initial data of spherically symmetric general relativity when spacetime is foliated extrinsically. When the dominant energy condition is satisfied these conditions assume a particularly simple form. Let $\rho_{Max}$ be the maximum value of the energy density and $\ell$ the radial measure of its support. If $\rho_{Max}\ell^2$ is bounded from above by some numerical constant, the initial data cannot possess an apparent horizon. This constant does not depend sensitively on the gauge. An analogous inequality is obtained for singularities with some larger constant. The derivation exploits Poincar\'e type inequalities to bound integrals over certain spatial scalars. A novel approach to the construction of analogous necessary conditions for general initial data is suggested.Comment: 15 pages, revtex, to appear in Phys. Rev.

Directivity enhancement and deflection of the beam emitted from a photonic crystal waveguide via defect coupling

Cataloged from PDF version of article.We experimentally and numerically investigate the spatial distribution of the emission from a photonic crystal waveguide, coupled with defects, that are located at the output edge. Two defects that are located symmetrically enhance the directivity of the beam compared to that of a plain waveguide, as was reported in recently conducted theoretical work. We further demonstrate that a single defect deflects of the beam. By choosing the defect resonance that is close to the edge of the pass band of the waveguide, where the group velocity of the beam within the waveguide is slow, a significant amount of deflection can be achieved. (c) 2007 Optical Society of Americ

The Constraints in Spherically Symmetric General Relativity II --- Identifying the Configuration Space: A Moment of Time Symmetry

We continue our investigation of the configuration space of general relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We show that MS configurations satisfy both the positive quasi-local mass (QLM) theorem and its converse. We derive an analytical expression for the spatial metric in the neighborhood of a generic singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tensor. We show that if the energy density of matter is monotonically decreasing, the geometry cannot be singular. A supermetric on the configuration space which distinguishes between singular geometries and non-singular ones is constructed explicitly. Global necessary and sufficient criteria for the formation of trapped surfaces and singularities are framed in terms of inequalities which relate appropriate measures of the material energy content on a given support to a measure of its volume. The strength of these inequalities is gauged by exploiting the exactly solvable piece-wise constant density star as a template.Comment: 50 pages, Plain Tex, 1 figure available from the authors

Hamiltonian Frenet-Serret dynamics

The Hamiltonian formulation of the dynamics of a relativistic particle described by a higher-derivative action that depends both on the first and the second Frenet-Serret curvatures is considered from a geometrical perspective. We demonstrate how reparametrization covariant dynamical variables and their projections onto the Frenet-Serret frame can be exploited to provide not only a significant simplification of but also novel insights into the canonical analysis. The constraint algebra and the Hamiltonian equations of motion are written down and a geometrical interpretation is provided for the canonical variables.Comment: Latex file, 14 pages, no figures. Revised version to appear in Class. Quant. Gra

Hamilton's equations for a fluid membrane: axial symmetry

Consider a homogenous fluid membrane, or vesicle, described by the Helfrich-Canham energy, quadratic in the mean curvature. When the membrane is axially symmetric, this energy can be viewed as an `action' describing the motion of a particle; the contours of equilibrium geometries are identified with particle trajectories. A novel Hamiltonian formulation of the problem is presented which exhibits the following two features: {\it (i)} the second derivatives appearing in the action through the mean curvature are accommodated in a natural phase space; {\it (ii)} the intrinsic freedom associated with the choice of evolution parameter along the contour is preserved. As a result, the phase space involves momenta conjugate not only to the particle position but also to its velocity, and there are constraints on the phase space variables. This formulation provides the groundwork for a field theoretical generalization to arbitrary configurations, with the particle replaced by a loop in space.Comment: 11 page

ADM Worldvolume Geometry

We describe the dynamics of a relativistic extended object in terms of the geometry of a configuration of constant time. This involves an adaptation of the ADM formulation of canonical general relativity. We apply the formalism to the hamiltonian formulation of a Dirac-Nambu-Goto relativistic extended object in an arbitrary background spacetime.Comment: 4 pages, Latex. Uses espcrc2.sty To appear in the proceedings of the Third Conference on Constrained Dynamics and Quantum Gravity, September, 1999. To appear in Nuclear Physics B (Proceedings Supplement